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### A dichotomy theorem for mono-unary algebras

Fundamenta Mathematicae

We study the isomorphism relation of invariant Borel classes of countable mono-unary algebras and prove a strong dichotomy theorem.

### A duality between unary algebras and their subuniverse lattices

Portugaliae mathematica

### A note on normal varieties of monounary algebras

Czechoslovak Mathematical Journal

A variety is called normal if no laws of the form $s=t$ are valid in it where $s$ is a variable and $t$ is not a variable. Let $L$ denote the lattice of all varieties of monounary algebras $\left(A,f\right)$ and let $V$ be a non-trivial non-normal element of $L$. Then $V$ is of the form $\mathrm{M}od\left({f}^{n}\left(x\right)=x\right)$ with some $n>0$. It is shown that the smallest normal variety containing $V$ is contained in $\mathrm{H}SC\left(\mathrm{M}od\left({f}^{mn}\left(x\right)=x\right)\right)$ for every $m>1$ where $\mathrm{C}$ denotes the operator of forming choice algebras. Moreover, it is proved that the sublattice of $L$ consisting of all normal elements of...

### Algebraic approach to locally finite trees with one end

Mathematica Bohemica

Let $T$ be an infinite locally finite tree. We say that $T$ has exactly one end, if in $T$ any two one-way infinite paths have a common rest (infinite subpath). The paper describes the structure of such trees and tries to formalize it by algebraic means, namely by means of acyclic monounary algebras or tree semilattices. In these algebraic structures the homomorpisms and direct products are considered and investigated with the aim of showing, whether they give algebras with the required properties. At...

### Antiatomic retract varieties of monounary algebras

Czechoslovak Mathematical Journal

### Atomary tolerances on finite algebras

Mathematica Bohemica

A tolerance on an algebra is defined similarly to a congruence, only the requirement of transitivity is omitted. The paper studies a special type of tolerance, namely atomary tolerances. They exist on every finite algebra.

### Bijective reflexions and coreflexions of commutative unars

Acta Universitatis Carolinae. Mathematica et Physica

### Cardinal arithmetic of a certain class of monounary algebras

Časopis pro pěstování matematiky

### Cardinalities of lattices of topologies of unars and some related topics

Discussiones Mathematicae - General Algebra and Applications

In this paper we find cardinalities of lattices of topologies of uncountable unars and show that the lattice of topologies of a unar cannor be countably infinite. It is proved that under some finiteness conditions the lattice of topologies of a unar is finite. Furthermore, the relations between the lattice of topologies of an arbitrary unar and its congruence lattice are established.

### Cardinality of retracts of monounary algebras

Czechoslovak Mathematical Journal

For an uncountable monounary algebra $\left(A,f\right)$ with cardinality $\kappa$ it is proved that $\left(A,f\right)$ has exactly ${2}^{\kappa }$ retracts. The case when $\left(A,f\right)$ is countable is also dealt with.

### Characterizations of certain general and reproductive solutions of arbitrary equations

Matematički Vesnik

### Characterizations of certain monounary algebras. I

Archivum Mathematicum

### Characterizations of certain monounary algebras. II

Archivum Mathematicum

### Church-Rosser property and decidability of monadic theories of unary algebras

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

### Congruence lattices of intransitive G-Sets and flat M-Sets

Commentationes Mathematicae Universitatis Carolinae

An M-Set is a unary algebra $〈X,M〉$ whose set $M$ of operations is a monoid of transformations of $X$; $〈X,M〉$ is a G-Set if $M$ is a group. A lattice $L$ is said to be represented by an M-Set $〈X,M〉$ if the congruence lattice of $〈X,M〉$ is isomorphic to $L$. Given an algebraic lattice $L$, an invariant $\Pi \left(L\right)$ is introduced here. $\Pi \left(L\right)$ provides substantial information about properties common to all representations of $L$ by intransitive G-Sets. $\Pi \left(L\right)$ is a sublattice of $L$ (possibly isomorphic to the trivial lattice), a $\Pi$-product lattice. A $\Pi$-product...

### Construction of all homomorphisms of groupoids

Czechoslovak Mathematical Journal

### Construction of all homomorphisms of mono-$n$-ary algebras

Czechoslovak Mathematical Journal

### Construction of all strong homomorphisms of binary structures

Czechoslovak Mathematical Journal

### Convex automorphisms of partial monounary algebras

Czechoslovak Mathematical Journal

### Convex subsets of partial monounary algebras

Czechoslovak Mathematical Journal

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